Seems like a faucet bushing would be threaded in. With a cross cut tooling notch to remove it.

A pressed in bushing, like in a flywheel use a Slide Hammer Puller. I usually rent/onloan one from the autopart store. I have even made a puller out of a carrage bolt on a grinder. ground the bolt head to fit and grab & a washer on a socket to pull into.